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I'm studying the concept of field applied to modular aritmetic. Is it correct to say that, if the dimension is a prime number $p$ then field properties are satisfied for the integers $\bmod p$ ? And that, if the dimension is a power of a prime number $p$ (that is $p^{n}$), not all integers $\bmod p^{n}$ form a field, but only a set of $p^{n}$ integers ?

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The integers modulo $p$ for a prime $p$ does form a field.

The integers modulo $p^n$ for $n>1$ doesn't form a field. And there isn't a subset of that which forms a field either.

There are fields with $p^n$ elements for any prime $p$ and natural number $n$. However, for $n>1$ this field does not coincide with the ring of integers modulo $p^n$. For instance, the additive group of such a field is $(\Bbb Z_p)^n$.

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  • $\begingroup$ Thanks so much. I would like to study your assertions on a book or websites, do you have some links? $\endgroup$
    – AleWolf
    May 13, 2019 at 19:30
  • $\begingroup$ @AleQuercia Any introductory book in abstract algebra (as long as it covers fields at all) should do. $\endgroup$
    – Arthur
    May 13, 2019 at 19:34
  • $\begingroup$ Thanks so much. $\endgroup$
    – AleWolf
    May 13, 2019 at 19:38
  • $\begingroup$ However, take a look at Theorem 4.1 here $\endgroup$
    – AleWolf
    May 13, 2019 at 19:59
  • $\begingroup$ @AleQuercia That seems to agree with me: The integers modulo $p$ is a field iff $p$ is prime. That's what theorem 4.1 says too. $\endgroup$
    – Arthur
    May 13, 2019 at 20:00

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